| L(s) = 1 | + 2·2-s + 2·4-s + 4·8-s − 6·9-s + 8·16-s − 12·18-s − 24·19-s + 4·29-s + 24·31-s + 8·32-s − 12·36-s + 24·37-s − 48·38-s − 2·49-s − 8·53-s + 8·58-s + 48·62-s + 8·64-s − 24·72-s + 48·74-s − 48·76-s + 9·81-s + 48·83-s − 4·98-s − 48·103-s − 16·106-s + 36·109-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.41·8-s − 2·9-s + 2·16-s − 2.82·18-s − 5.50·19-s + 0.742·29-s + 4.31·31-s + 1.41·32-s − 2·36-s + 3.94·37-s − 7.78·38-s − 2/7·49-s − 1.09·53-s + 1.05·58-s + 6.09·62-s + 64-s − 2.82·72-s + 5.57·74-s − 5.50·76-s + 81-s + 5.26·83-s − 0.404·98-s − 4.72·103-s − 1.55·106-s + 3.44·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.656899299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.656899299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 154 T^{2} + 20859 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83957040714112867614551492601, −7.17139850154194940540557527771, −6.67450551672930849786619109366, −6.60897355234665664644116675271, −6.51939265017407653763431927850, −6.32267782374629733809024737040, −6.00145135498016421227176469775, −5.93118694838157245823471270744, −5.89901175335517531165153752718, −5.26038683804315955313386208851, −4.88968005894662091227957285750, −4.68232537050479925388101802253, −4.63197027447726963819828443750, −4.53590527265417600062382932324, −3.96125512411985533405767238561, −3.93191817566789420835583005662, −3.84819847360057437746789104349, −2.99429481514725272822024367170, −2.80762299733371432146553395012, −2.64100216716250976032637645858, −2.55836021125705032684102506303, −1.90697823035968377339251657801, −1.89038467419989147946381234588, −0.899508843648121097689387714738, −0.53070849022740900579759320909,
0.53070849022740900579759320909, 0.899508843648121097689387714738, 1.89038467419989147946381234588, 1.90697823035968377339251657801, 2.55836021125705032684102506303, 2.64100216716250976032637645858, 2.80762299733371432146553395012, 2.99429481514725272822024367170, 3.84819847360057437746789104349, 3.93191817566789420835583005662, 3.96125512411985533405767238561, 4.53590527265417600062382932324, 4.63197027447726963819828443750, 4.68232537050479925388101802253, 4.88968005894662091227957285750, 5.26038683804315955313386208851, 5.89901175335517531165153752718, 5.93118694838157245823471270744, 6.00145135498016421227176469775, 6.32267782374629733809024737040, 6.51939265017407653763431927850, 6.60897355234665664644116675271, 6.67450551672930849786619109366, 7.17139850154194940540557527771, 7.83957040714112867614551492601
